Our construction of cumulant ratios is similar to that of the Binder
cumulants [1]}, [2]}, for this reason, we will use the
term geometric Binder cumulant (GBC). The GBCs are zero if the
adiabatic path encounters no gap closure points, otherwise they are
finite. As a preliminary example, we calculate the GBC for a
spin-\(\frac{1}{2}\)  particle in a precessing magnetic field. We then
implement our construction in the
MTP [3]}, [4]}, [5]}, [6]}, [7]} and
show that the GBCs are an effective tool for finite size scaling of
metal-insulator [8]} and other quantum phase transitions.
Gap closure points (or regions) can be located. As a technicality, we
emphasize the use of a particular application of finite difference
derivatives which guarantee the correct scaling with system size in
gapless systems.
